1. Field of the Invention
The present invention relates to a geographic map with a homogeneous grid system on a flat or curved support.
2. Description of the Prior Art
It is known that the earth, which may be assimilated in a first approximation with a sphere of radius R, is normally represented in plane images in accordance with a considerable number of methods derived from formulations of the general type:
XP=f(l,L) PA1 YP=g(l,L)
where XP and YP are the plane coordinates, l is the longitude and L is the latitude.
In order to facilitate map plotting and referencing, a sphere grid network is also superimposed on the map. The grid system or network in most common use corresponds to lines of constant longitude and of constant latitude (meridians, parallels) which make it possible to obtain coordinates usually designated as the geographic coordinates.
Depending on the nature of the functions (f, g), it is possible to have representations (commonly but erroneously designated as projections) which are either conformal (preservation of infinitesimal angles), equivalent (preservation of areas) or aphylactic (in which neither angles nor areas are retained).
For the purpose of map coverage of a region having a small area, it is a customary practice to define a single projection in which there are sometimes formed a number of connectable interruptions or breaks, the complete assembly being intended to constitute a continuous representation (without either duplication or tearing) of the zone to be represented.
It is possible to construct a single projection for the whole world but in this case certain portions are inadequately represented: in particular, a single conformal projection cannot represent the entire world without singularity.
In the case of large areas and all the more so in the case of the whole world, it is a common practice to construct a projective system or in other words a family of projections (in which the functions (f, g) differ only in respect of numerical coefficients), which cover the region to be represented but with lines on which the interruptions or breaks of two adjacent projections do not fit together.
The centers of the projections are adjusted by virtue of considerations which are very different but are often determined as a function of the geographic coordinates.
Among the representations, conformal projections are in very wide use since they represent the terrain in a similitude of ratio K (scale) which, as a general rule, is stationary on a central element (line or point) and increases parabolically as the distance from said central element is greater.
Plotting in a geographic coordinate system (meridians, parallels) is fairly satisfactory locally, (in particular at the equator) but becomes singular in the polar regions. The meridians in fact converge in the polar regions and the longitude becomes indeterminate. Furthermore, the areas delimited by meridians and parallels are relatively rectangular in the vicinity of the equator and become triangular in the vicinity of the poles, with the result that the subdivisions correspond to elements having very variable areas or else to elements having very different shapes if the procedure adopted consists in regrouping.
It has already been sought to establish on the terrestrial sphere grid systems forming substantially equal areas, these areas being intended to permit subdivision into grid meshes having areas which are in turn substantially equal irrespective of the part of the globe considered.
Among the methods proposed up to the present time, the method which offers the highest degree of fineness appears to have been given by Popko (1968) (ref.: G. H. Dutton: Geodesic modeling of planetary relief--Cartographica--Auto Carto Six--Selected papers--Volume 21/Numbers 2 and 3; Sixth International Symposium on Automated Cartography Ottawa--Hull, October 16-21, 1983. University of Toronto Press--1984; pages 192-193) which has proposed, among others, the subdivision of a grid system of sixty equal spherical isosceles triangles obtained for example by gnomonic transfer to the sphere of the edges of the semiregular polyhedron of the second type which is circumscribed about the sphere and has sixty equal triangular faces.
Starting from the initial grid system, it is proposed to subdivide each triangle into four spherical triangles by joining the midpoints of the sides by three great-circle arcs, whereupon each triangle can subsequently be iteratively subdivided in accordance with the same method. However, in each subdivision, the area of the central triangle is larger than the area of the end triangles. Furthermore, the method is not very simple from a numerical standpoint.